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A218702
Number of partitions of n in which any two distinct parts differ by at least 7.
2
1, 1, 2, 2, 3, 2, 4, 2, 4, 4, 6, 6, 11, 10, 13, 16, 19, 20, 25, 27, 33, 34, 39, 41, 51, 52, 61, 65, 80, 82, 99, 104, 126, 133, 156, 168, 199, 209, 243, 261, 302, 320, 372, 392, 447, 479, 537, 572, 650, 693, 770, 829, 920, 982, 1096, 1169, 1306, 1396, 1541
OFFSET
0,3
COMMENTS
Also number of partitions of n in which each part, with the possible exception of the largest, occurs at least 7 times.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
FORMULA
G.f.: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(7*i)/(1-x^i)).
log(a(n)) ~ sqrt((2*Pi^2/3 + 4*c)*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-7*x)) dx = -1.104868234083422137620242346741601264555358762... - Vaclav Kotesovec, Jan 28 2022
EXAMPLE
a(8) = 4: [1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [8].
a(9) = 4: [1,1,1,1,1,1,1,1,1], [3,3,3], [1,8], [9].
a(10) = 6: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [1,1,8], [1,9], [10].
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +add(b(n-i*j, i-7), j=1..n/i)))
end:
a:= n-> b(n, n):
seq(a(n), n=0..70);
CROSSREFS
Column k=7 of A218698.
Cf. A160977.
Sequence in context: A001616 A257599 A366620 * A363525 A324181 A297169
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Nov 04 2012
STATUS
approved